Defected Temporal Graphs (DTGs)

• Defected Temporal Graphs are specialized temporal constructs that certify contraction in time-varying Friedkin–Johnsen opinion dynamics by encoding propagated row-sum defects from stubborn agents.
• They quantify stability by enforcing ε-stubborn conditions and leveraging influential paths to impose explicit bounds on the state-transition matrix.
• DTGs differ from weakly defected temporal graphs by providing precise quantitative contraction certificates that guarantee convergence in evolving opinion systems.

Defected Temporal Graphs (DTGs) are temporal graph structures introduced as graph-theoretic certificates for stability of time-varying Friedkin–Johnsen opinion dynamics. In this setting, temporal connectivity is not studied as an end in itself: the decisive question is whether stubborn influence repeatedly reaches all agents through temporally valid paths, so that the homogeneous state-transition matrix becomes contractive. A DTG encodes this quantitative condition, while a Weakly Defected Temporal Graph (WDTG) encodes a weaker qualitative variant; together they connect temporal graph structure, products of nonnegative substochastic matrices, and the long-run behavior of evolving opinion systems (Abedinzadeh et al., 2 Oct 2025).

1. Conceptual position within temporal graph theory

Temporal graphs are commonly formalized either as event-expanded graphs built from time-stamped node instances and waiting edges, or as sequences of snapshots such as $G=(V,E_0,E_1,\ldots,E_{\tau-1})$ (0807.2357, Bui-Xuan et al., 12 Feb 2026). DTGs belong to this broader temporal-graph landscape, but they are not a generic representation class of the form “any graph with defects.” They arise inside a specific dynamical-systems problem: the stability analysis of time-varying Friedkin–Johnsen dynamics (Abedinzadeh et al., 2 Oct 2025).

The word “defected” is therefore specialized. In the DTG framework, the relevant defect is a row-sum defect in the matrix $\Lambda[t]W[t]$, caused by stubbornness. If agent $i$ satisfies $\lambda_i[t]<1$, then the $i$-th row of $\Lambda[t]W[t]$ sums to less than one, and that deficiency can propagate through time along influence paths. A DTG is precisely a temporal graph window in which this propagated defect reaches all agents strongly enough to imply contraction of the state-transition product (Abedinzadeh et al., 2 Oct 2025).

This meaning should be distinguished from several neighboring uses of “defect” in temporal-graph research. In some work, defects refer to data inconsistencies or rule violations; in others, to temporal-orientation obstructions or to damage/noise in predictive graph learning. DTGs, in the strict sense, are instead contraction certificates tied to stubborn influence in opinion dynamics. This positions the notion closer to stability theory than to anomaly detection, graph cleaning, or temporal representation learning (Abedinzadeh et al., 2 Oct 2025).

2. Formal model, path structure, and definitions

The underlying dynamics are the time-varying Friedkin–Johnsen model over agents

$\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$

with expressed opinions $\mathbf{x}[t]$ and innate opinions $\mathbf{s}$. The update equation is

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$

where $\Lambda[t]W[t]$0 is row-stochastic and

$\Lambda[t]W[t]$1

The state-transition matrix is

$\Lambda[t]W[t]$2

The system is asymptotically stable if $\Lambda[t]W[t]$3, and exponentially stable if there exist $\Lambda[t]W[t]$4, $\Lambda[t]W[t]$5 such that

$\Lambda[t]W[t]$6

(Abedinzadeh et al., 2 Oct 2025).

At each time $\Lambda[t]W[t]$7, the influence matrix induces a directed graph

$\Lambda[t]W[t]$8

with

$\Lambda[t]W[t]$9

Thus $i$0 means that agent $i$1 influences agent $i$2. Over an interval $i$3, the temporal graph is

$i$4

A temporal edge is the triplet $i$5, and an edge is a $i$6-edge if $i$7 for some threshold $i$8 (Abedinzadeh et al., 2 Oct 2025).

The definitions of stubbornness are central. Agent $i$9 is stubborn at time $\lambda_i[t]<1$0 if

$\lambda_i[t]<1$1

and strictly stubborn or $\lambda_i[t]<1$2-stubborn if there exists $\lambda_i[t]<1$3 such that

$\lambda_i[t]<1$4

A temporal path starting at a stubborn agent is an $\lambda_i[t]<1$5-path; a temporal path starting at an $\lambda_i[t]<1$6-stubborn agent and using only $\lambda_i[t]<1$7-edges is an influential-path (Abedinzadeh et al., 2 Oct 2025).

The two key graph notions are then defined as follows.

Notion	Requirement on every agent in a window $\lambda_i[t]<1$8	Strength
WDTG	stubborn, or connected to a stubborn agent via a finite $\lambda_i[t]<1$9-path	qualitative
DTG	$i$0-stubborn, or connected to an $i$1-stubborn agent via a finite influential-path	quantitative

More precisely, a temporal graph $i$2 is a WDTG if there exists $i$3 such that, in layer $i$4, every agent is either stubborn or connected to a stubborn agent via a finite $i$5-path fully contained in $i$6. It is a DTG if there exists $i$7 such that, in layer $i$8, every agent is either $i$9-stubborn or connected to an $\Lambda[t]W[t]$0-stubborn agent via a finite influential-path entirely within $\Lambda[t]W[t]$1 (Abedinzadeh et al., 2 Oct 2025).

Two structural assumptions are also imposed. Assumption 1 states

$\Lambda[t]W[t]$2

and Assumption 2 excludes any time $\Lambda[t]W[t]$3 such that $\Lambda[t]W[t]$4 (Abedinzadeh et al., 2 Oct 2025).

3. DTGs as quantitative contraction certificates

The main technical role of a DTG is to certify strict contraction of the homogeneous dynamics. The fundamental lemma states that if $\Lambda[t]W[t]$5 is a DTG, then

$\Lambda[t]W[t]$6

This inequality is the core quantitative content of the definition: the window is not merely connected to stubborn agents, but connected strongly enough to force a uniform row-sum contraction (Abedinzadeh et al., 2 Oct 2025).

The proof mechanism is structural. If an agent is itself $\Lambda[t]W[t]$7-stubborn, then its row sum is at most $\Lambda[t]W[t]$8. If an agent is reached from an $\Lambda[t]W[t]$9-stubborn source by an influential-path of length $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$0, then at least $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$1 of the upstream defect propagates to that row, yielding a bound of the form

$\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$2

Taking the maximal path length over agents produces the uniform estimate above (Abedinzadeh et al., 2 Oct 2025).

This immediately extends to unions of DTG windows. If a time interval is partitioned into consecutive subintervals and $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$3 of them are DTGs with lengths $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$4, then

$\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$5

Hence repeated DTG windows multiply strict contraction factors (Abedinzadeh et al., 2 Oct 2025).

From this lemma, the principal stability results follow. If there exist infinitely many pairwise-disjoint finite intervals $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$6 such that each temporal graph $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$7 is a DTG, then the TVFJ system is asymptotically stable. If, more strongly, every sliding window of fixed length $\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$8 is defected—what the paper calls a semi-periodic defected network—then the system is exponentially stable with

$\mathcal{V}=\{\mathrm{v}_1,\ldots,\mathrm{v}_n\},$9

(Abedinzadeh et al., 2 Oct 2025).

WDTGs behave differently. If an interval contains at least one WDTG, then

$\mathbf{x}[t]$0

but no explicit quantitative factor like $\mathbf{x}[t]$1 is available. This distinction is essential. The paper gives a counterexample for general TVFJ using

$\mathbf{x}[t]$2

where each step is a WDTG, yet

$\mathbf{x}[t]$3

Thus WDTG recurrence alone does not guarantee asymptotic stability in the general model (Abedinzadeh et al., 2 Oct 2025).

4. Trust-based extension, omega-limit structure, and robustness

The paper also studies a trust-based Friedkin–Johnsen extension in which the time variation is structured by a fixed trust matrix $\mathbf{x}[t]$4 and a time-varying adjacency $\mathbf{x}[t]$5. The weights are defined by

$\mathbf{x}[t]$6

and susceptibility is determined by a neighborhood function

$\mathbf{x}[t]$7

In this structured setting, infinitely many disjoint WDTG intervals of uniformly bounded length are sufficient for asymptotic stability (Abedinzadeh et al., 2 Oct 2025).

The reason is combinatorial finiteness. Because $\mathbf{x}[t]$8, the number of possible adjacency matrices is finite. Given fixed $\mathbf{x}[t]$9, each adjacency determines a unique $\mathbf{s}$0, and since each $\mathbf{s}$1 depends only on $\mathbf{s}$2, each adjacency also determines $\mathbf{s}$3. With bounded WDTG interval length, only finitely many WDTG block types can occur. At least one must recur infinitely often, and each such block contracts strictly, which suffices for asymptotic stability (Abedinzadeh et al., 2 Oct 2025).

Beyond stability, the paper characterizes long-run behavior. The solution admits the representation

$\mathbf{s}$4

Defining

$\mathbf{s}$5

the paper proves that $\mathbf{s}$6 is row-substochastic for all $\mathbf{s}$7. When the system is asymptotically stable, every accumulation point satisfies

$\mathbf{s}$8

Thus the omega-limit set is contained in the convex hull of innate beliefs (Abedinzadeh et al., 2 Oct 2025).

For periodically switching systems of period $\mathbf{s}$9, the dynamics can be decomposed into a $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$0-LTI family

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$1

with

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$2

and

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$3

If $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$4 is a WDTG, then the system is exponentially stable and the omega-limit set contains at most $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$5 points,

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$6

The bound $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$7 is explicit and tight in the theorem’s formulation (Abedinzadeh et al., 2 Oct 2025).

Robustness is addressed through a perturbed system

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$8

with

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$9

If the nominal model is exponentially stable with constants $\Lambda[t]W[t]$00 and $\Lambda[t]W[t]$01, and

$\Lambda[t]W[t]$02

then the perturbed dynamics remain exponentially stable. DTGs matter here because they are what generate the nominal exponential-stability constants in the first place (Abedinzadeh et al., 2 Oct 2025).

5. Relation to adjacent “defect” notions in temporal-graph research

In neighboring temporal-graph subfields, related terms refer to substantially different objects. In model checking on temporal graphs, the closest construction is the differential

$\Lambda[t]W[t]$03

defined as the static expansion graph of a sliding window of $\Lambda[t]W[t]$04 consecutive snapshots. That derivative-like object is explicitly not a difference graph of edge additions and removals, and it is not a DTG in the Friedkin–Johnsen sense; it is instead a bounded-window static expansion used for width measures and local FO/MSO reasoning (Bui-Xuan et al., 12 Feb 2026).

In temporal comparability theory, “defects” correspond to violations of temporal transitivity, correlated monolabel triangles, contradiction patterns in implication digraphs, or cycles in the set of necessary arcs. There, the main obstruction is temporal orientability rather than contraction of a state-transition matrix. A graph is defective when its time-label/orientation constraints cannot be completed into a temporal transitive orientation, not when stubborn influence fails to propagate (Charbit et al., 8 Oct 2025).

In temporal graph data quality, the closest analogue to a defected temporal graph is a graph violating a set of Temporal Graph Functional Dependencies. A TGFD

$\Lambda[t]W[t]$05

declares a structural-temporal consistency rule, and the defect set is the violation set

$\Lambda[t]W[t]$06

This notion of defect is rule-theoretic and data-centric: it concerns inconsistent matched subgraphs across snapshots, not contraction or stability (Alipourlangouri et al., 2021).

In temporal graph learning, discrete-time dynamic graphs are typically modeled as sequences

$\Lambda[t]W[t]$07

with downstream tasks such as future link prediction. That literature can discuss noisy, partially observed, sparsified, or structurally damaged graphs, but it does not define DTGs as graph-theoretic stability certificates. The emphasis there is representation learning, sequence encoding, and pairwise prediction rather than stubbornness-induced contraction (Chen et al., 2024).

These contrasts clarify a common misconception. “Defected temporal graph” is not a universal synonym for corrupted temporal data, anomalous interaction streams, or temporal inconsistency in general. In the strict technical sense established in the opinion-dynamics literature, a DTG is a temporal window certifying propagated row-sum defect and hence contraction (Abedinzadeh et al., 2 Oct 2025).

6. Scope, limitations, and broader significance

The DTG framework is strong precisely because it is specialized. It converts a stability question for a linear time-varying opinion process into a graph condition that is interpretable: repeated reachability from stubborn agents through time-respecting influence chains. It is also quantitatively explicit, since the contraction rate depends on $\Lambda[t]W[t]$08, $\Lambda[t]W[t]$09, and the window length $\Lambda[t]W[t]$10 or $\Lambda[t]W[t]$11 (Abedinzadeh et al., 2 Oct 2025).

The same specialization also marks its limits. The framework is built for time-varying Friedkin–Johnsen systems and their trust-based extension; it is not introduced as a general anomaly-detection formalism, a missing-data model, or a universal representation of damaged temporal networks. The paper emphasizes graph-based interpretability and scalability, but it does not provide a dedicated algorithmic complexity analysis for DTG recognition or extraction (Abedinzadeh et al., 2 Oct 2025).

A second limitation is the separation between DTGs and WDTGs. In general TVFJ, WDTGs are too weak: recurring qualitative access to stubborn agents does not prevent contraction factors from approaching one too quickly. Only the quantitative DTG condition yields the explicit bound

$\Lambda[t]W[t]$12

This sharp distinction is one of the central conceptual contributions of the theory (Abedinzadeh et al., 2 Oct 2025).

A broader implication is that temporal graph theory now contains several non-equivalent ways to formalize “defect”: contraction certificates in opinion dynamics, temporal-transitivity obstructions, rule violations in evolving graph data, and localized temporal derivatives for logic and model checking (Abedinzadeh et al., 2 Oct 2025, Charbit et al., 8 Oct 2025, Alipourlangouri et al., 2021, Bui-Xuan et al., 12 Feb 2026). This suggests that “defect” in temporal graphs is not a single primitive but a family of semantics indexed by task: dynamical stability, orientation consistency, data integrity, or local temporal structure.

A plausible implication is that future DTG research could combine these perspectives. The stability-oriented DTG notion already provides a rigorous bridge between temporal paths and nonnegative-matrix contraction. Rule-based TGFD frameworks provide explicit defect witnesses in data; derivative-based temporal expansions provide logical locality; and topological pipelines based on temporal motifs and persistent homology provide graph-level signatures of temporal irregularity (Pritam et al., 14 Feb 2025). Such an overview would move beyond the current, task-specific meanings of defect while preserving the mathematical precision that made DTGs useful in the first place.