Weakly Defected Temporal Graphs

• WDTGs are temporal networks defined by the existence of at least one layer where every agent is either stubborn or connected to a stubborn agent via an s-path, ensuring qualitative reachability.
• They certify strict yet nonuniform contraction of state-transition matrices in time-varying Friedkin–Johnsen models, thus aiding stability analysis.
• WDTGs offer a more general framework than defected temporal graphs by not requiring uniform lower bounds on stubbornness or influence edge weights.

Searching arXiv for the cited papers to ground the article in the latest records. Weakly defected temporal graphs (WDTGs) are temporal-network structures introduced in the analysis of time-varying Friedkin–Johnsen opinion dynamics as graph-theoretic certificates of how stubborn influence propagates through time. Formally, over a finite interval $[t_0,t_d)$, a temporal graph is a WDTG if there exists a layer $k\in[t_0,t_d)$ such that every agent is either stubborn in that layer or connected to a stubborn agent by a finite $s$-path fully contained in the interval. In this framework, WDTGs are weaker than defected temporal graphs (DTGs): they encode temporal reachability from stubborn agents, but do not impose uniform lower bounds on stubbornness or edge weights. Their principal role is to certify strict contraction of finite products of state-transition matrices and, under additional structure, asymptotic or exponential stability of evolving opinion systems (Abedinzadeh et al., 2 Oct 2025).

1. Formal definition and ambient dynamical system

The ambient model is the time-varying Friedkin–Johnsen (TVFJ) system on an agent set

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

with expressed opinions $x_i[t]\in[0,1]$ and fixed innate opinions $s_i\in[0,1]$. Writing

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$

the dynamics are

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

where $W[t]$ is row-stochastic and $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$ with $k\in[t_0,t_d)$0. An agent $k\in[t_0,t_d)$1 is stubborn at time $k\in[t_0,t_d)$2 if $k\in[t_0,t_d)$3, and is strictly stubborn, or $k\in[t_0,t_d)$4-stubborn, if there exists $k\in[t_0,t_d)$5 such that $k\in[t_0,t_d)$6 (Abedinzadeh et al., 2 Oct 2025).

At each time $k\in[t_0,t_d)$7, the influence matrix $k\in[t_0,t_d)$8 induces a directed graph

$k\in[t_0,t_d)$9

Thus an edge $s$0 means that agent $s$1 assigns positive influence weight to agent $s$2. Over a time interval $s$3, the temporal graph is

$s$4

viewed as a multiplex directed graph. A temporal edge is a triplet $s$5 whenever $s$6. If $s$7, the edge is called a $s$8-edge.

A temporal path from $s$9 to $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$0 over $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$1 is a sequence of temporal edges across consecutive time steps. The paper distinguishes two path types. An $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$2-path is a temporal path whose starting node is stubborn. An influential-path is a temporal path whose starting node is $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$3-stubborn and whose edges are all $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$4-edges.

With this notation, the exact WDTG definition is: $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$5 is a weakly defected temporal graph if there exists $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$6 such that, in layer $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$7, every agent $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$8 is either stubborn or connected to a stubborn agent via a finite $\mathcal{V}=\{\mathrm{v}_1,\dots,\mathrm{v}_n\},$9-path fully contained in $x_i[t]\in[0,1]$0. The time window, not any single snapshot, is therefore the object on which the definition acts.

2. WDTGs versus defected temporal graphs

The corresponding stronger notion is the defected temporal graph. A temporal graph $x_i[t]\in[0,1]$1 is a DTG if there exists $x_i[t]\in[0,1]$2 such that every agent is either $x_i[t]\in[0,1]$3-stubborn in layer $x_i[t]\in[0,1]$4 or connected to an $x_i[t]\in[0,1]$5-stubborn agent through a finite influential-path entirely within $x_i[t]\in[0,1]$6 (Abedinzadeh et al., 2 Oct 2025).

The distinction is structural and quantitative. A WDTG requires only stubbornness, $x_i[t]\in[0,1]$7, and temporal reachability by an $x_i[t]\in[0,1]$8-path. A DTG adds two uniform lower bounds: a susceptibility margin $x_i[t]\in[0,1]$9 and a path-edge threshold $s_i\in[0,1]$0. For that reason, every DTG is a WDTG, but the converse generally fails. A WDTG may have stubbornness values arbitrarily close to $s_i\in[0,1]$1 and arbitrarily small positive influence weights along the relevant temporal paths.

This difference is central to interpretation. A WDTG encodes a qualitative reachability statement: stubborn influence can, in principle, reach every node through time. A DTG encodes a stronger quantitative propagation statement: stubborn influence reaches every node with a uniformly bounded strength over the interval. The connectivity involved is temporal reachability over a finite window, not static connectivity of a single graph.

The same paper also states a time-invariant characterization. If $s_i\in[0,1]$2 and $s_i\in[0,1]$3 are constant, then the system is asymptotically stable if and only if the influence graph induced by $s_i\in[0,1]$4 forms a WDTG over some interval $s_i\in[0,1]$5 with $s_i\in[0,1]$6. In this setting, the WDTG criterion recovers the familiar condition that every node is either stubborn or influenced directly or indirectly by a stubborn node.

3. Matrix contraction and stability meaning

The solution of the TVFJ dynamics from time $s_i\in[0,1]$7 can be written as

$s_i\in[0,1]$8

where the state-transition matrix is

$s_i\in[0,1]$9

The norm used is the $$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$0-induced matrix norm,

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$1

Asymptotic stability means

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$2

while exponential stability means that there exist $$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$3 and $$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$4 such that

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$5

The matrix role of a WDTG is strict but nonuniform contraction. If $$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$6 contains at least one WDTG, then

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$7

This is weaker than the DTG estimate

$$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$8

which provides an explicit contraction gap from $$\mathbf{x}[t]=\begin{bmatrix}x_1[t]&\cdots&x_n[t]\end{bmatrix}^\top,\qquad \mathbf{s}=\begin{bmatrix}s_1&\cdots&s_n\end{bmatrix}^\top,$$9. For WDTGs, the contraction factor is only known to be strictly subunit; no time-uniform lower bound on the defect of row sums is supplied (Abedinzadeh et al., 2 Oct 2025).

A common misconception is that repeated strict contractions of the form $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$0 automatically imply asymptotic stability in the general TVFJ model. The paper provides a counterexample. Let

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

Then every layer is a WDTG, because all agents are stubborn at every time. However,

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

so $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$3. The example isolates the exact limitation of WDTGs in generic TVFJ systems: they guarantee topological access to stubborn influence, but not enough quantitative force to ensure decay of infinite matrix products.

4. Regimes in which WDTGs do imply stability

The strongest positive WDTG result in the source paper concerns the trust-based Friedkin–Johnsen (TBFJ) extension. In that model, a time-varying adjacency matrix $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$4 and a fixed row-stochastic trust matrix $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$5 determine

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

and susceptibilities are given by local functions

$$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$7

The dynamics retain the same TVFJ form. In this trust-based setting, if there exist infinitely many disjoint finite intervals $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$8 such that each $$\mathbf{x}[t+1]=\Lambda[t]W[t]\mathbf{x}[t]+(I-\Lambda[t])\mathbf{s},$$9 is a WDTG and

$W[t]$0

for some finite $W[t]$1, then the system is asymptotically stable (Abedinzadeh et al., 2 Oct 2025).

The mechanism is combinatorial rather than purely metric. Because $W[t]$2, only finitely many adjacency matrices can occur. In TBFJ, each adjacency matrix uniquely determines $W[t]$3 through $W[t]$4 and $W[t]$5 through the local neighborhood functions $W[t]$6. Uniformly bounded WDTG window length therefore implies that only finitely many distinct WDTG interval types can appear. Since each such type gives some contraction factor $W[t]$7, at least one type must recur infinitely often, and repeated multiplication by that same $W[t]$8 forces $W[t]$9.

A second regime is periodic switching. For a periodically switching FJ system with period $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$0,

$$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$1

if the period block $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$2 is a WDTG, then the system is exponentially stable with

$$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$3

The same result yields a $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$4-LTI decomposition and the bound

$$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$5

for the $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$6-limit set, whose points are

$$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$7

Here the periodic repetition of a single WDTG block converts strict one-block contraction into uniform repeated contraction.

5. Long-run behavior, boundedness, and robustness

The paper also analyzes the forcing term

$$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$8

showing that $$\Lambda[t]=\mathrm{diag}(\lambda_1[t],\dots,\lambda_n[t])$$9 is row-substochastic for all $k\in[t_0,t_d)$00 (Abedinzadeh et al., 2 Oct 2025). Consequently,

$k\in[t_0,t_d)$01

is always a combination of the initial condition and a row-substochastic combination of innate beliefs. Whenever asymptotic stability has been established, the initial-condition term vanishes and every accumulation point satisfies

$k\in[t_0,t_d)$02

Thus the $k\in[t_0,t_d)$03-limit set remains inside the convex hull of innate opinions. This boundedness conclusion does not require WDTGs directly; WDTGs enter by providing sufficient conditions for asymptotic or exponential stability in the appropriate structured settings.

The robustness result is formulated for exponentially stable nominal systems. The perturbed dynamics are

$k\in[t_0,t_d)$04

with nonnegative $k\in[t_0,t_d)$05, diagonal nonnegative $k\in[t_0,t_d)$06, and

$k\in[t_0,t_d)$07

Assume

$k\in[t_0,t_d)$08

where $k\in[t_0,t_d)$09 is row-stochastic, $k\in[t_0,t_d)$10 is diagonal with entries in $k\in[t_0,t_d)$11, and $k\in[t_0,t_d)$12 is uniformly bounded. If the nominal system is exponentially stable with constants $k\in[t_0,t_d)$13, and

$k\in[t_0,t_d)$14

then the perturbed system is also exponentially stable. A direct implication is that WDTG-based exponential stability in the periodic switching regime persists under sufficiently small bounded perturbations. The paper does not state an analogous general robustness theorem for the asymptotic-stability result based on recurring bounded-length WDTGs in TBFJ.

The assumptions attached to the TVFJ framework are also material. The paper requires susceptibility–row consistency,

$k\in[t_0,t_d)$15

and excludes the trivial case in which there exists a time $k\in[t_0,t_d)$16 with $k\in[t_0,t_d)$17.

6. Relation to other temporal-graph formalisms

The WDTG notion belongs to an opinion-dynamics setting and should be distinguished from other uses of temporal defects in the broader temporal-graph literature. One nearby construction is the weighted temporal event graph, a static directed acyclic graph $k\in[t_0,t_d)$18 built from a temporal network $k\in[t_0,t_d)$19 by taking events as vertices, temporal adjacencies as directed edges, and inter-event times as weights: $k\in[t_0,t_d)$20 This representation preserves time-respecting paths and timing differences between consecutive events and yields a thresholded subgraph $k\in[t_0,t_d)$21 for $k\in[t_0,t_d)$22-constrained paths (Saramäki et al., 2019). The source does not define WDTGs, but the event-graph framework isolates a related design space: temporal admissibility is encoded by path rules and edge weights. This suggests a useful comparison, not a formal equivalence. WDTGs certify propagation of stubborn influence across temporal layers, whereas weighted event graphs encode temporal path structure itself.

A second adjacent line of work studies $k\in[t_0,t_d)$23-edge-deficient temporal graphs, in which each snapshot differs from a fixed graph or spanning-tree backbone by at most $k\in[t_0,t_d)$24 missing edges. For always-connected $k\in[t_0,t_d)$25-edge-deficient temporal graphs on $k\in[t_0,t_d)$26 vertices, temporal exploration is possible in

$k\in[t_0,t_d)$27

steps, with $k\in[t_0,t_d)$28 exploration time when $k\in[t_0,t_d)$29 is constant (Lahtin et al., 15 May 2026). This is a defect model of bounded edge deletion relative to a near-static reference structure. It is conceptually comparable to the phrase “weakly defected,” but it is not the same formal object as a WDTG in the TVFJ sense. In particular, the opinion-dynamics WDTG is defined through stubborn nodes and temporal influence paths, not through per-snapshot deviation from a fixed backbone.

Within the literature represented here, WDTGs therefore occupy a specific niche. They are not a generic synonym for sparse temporal damage, nor a purely topological replacement for matrix analysis. Rather, they are a graph-theoretic certificate that links temporal reachability of stubborn influence to contraction properties of products of $k\in[t_0,t_d)$30. Their power is strongest when the temporal system has additional structure—especially fixed trust architecture, bounded-length recurrence, or exact periodic repetition—and correspondingly limited in the fully general time-varying setting (Abedinzadeh et al., 2 Oct 2025).