When not to target negative ties? Studying competitive influence maximisation in signed networks

TL;DR

This work extends influence maximisation to signed networks by incorporating negative ties into competitive voter dynamics, introducing a sign-aware optimization (GA) and a sign-agnostic baseline (GA+). It derives a matrix-analytic steady-state framework $X_A^* = \frac{1}{N}\mathbf{1}^T [L+\operatorname{diag}(p_A+p_B)]^{-1}(p_A-\mathbf{1}^T W^-)$ and optimises $p_A$ via gradient ascent, with MF analytical support and semi-analytical solutions for structured networks. Empirical validation shows real-world gains of up to ~9% in a Bitcoin OTC network and larger gains (up to ~18% in certain topology/resource regimes) when negative ties are strategically exploited, though game-theoretic analysis reveals scenarios where knowledge of negative ties can backfire. The results demonstrate the nontrivial benefits and strategic caveats of modeling negative ties in influence campaigns, with implications for public health, politics, and marketing in complex social networks.

Abstract

We explore the influence maximisation problem in networks with negative ties. Where prior work has focused on unsigned networks, we investigate the need to consider negative ties in networks while trying to maximise spread in a population - particularly under competitive conditions. Given a signed network we optimise the strategies of a focal controller, against competing influence in the network, using two approaches - either the focal controller uses a sign-agnostic approach or they factor in the sign of the edges while optimising their strategy. We compare the difference in vote-shares (or the share of population) obtained by both these methods to determine the need to navigate negative ties in these settings. More specifically, we study the impact of: (a) network topology, (b) resource conditions and (c) competitor strategies on the difference in vote shares obtained across both methodologies. We observe that gains are maximum when resources available to the focal controller are low and the competitor avoids negative edges in their strategy. Conversely, gains are insignificant irrespective of resource conditions when the competitor targets the network indiscriminately. Finally, we study the problem in a game-theoretic setting, where we simultaneously optimise the strategies of both competitors. Interestingly we observe that, strategising with the knowledge of negative ties can occasionally also lead to loss in vote-shares.

Figures (8)

• Figure 1: Simulations are performed on the single-largest connected component ($N = 4734$ nodes) of the Bitcoin OTC network. Panel (a) shows step-wise allocations to nodes in the $GA$ algorithm. The axes represent the total allocations $\alpha$ to nodes with negative links and the total allocations $\beta$ to nodes with strictly positive edges. Panel (b) shows the change in vote-shares ($X_{A}$) obtained by both algorithms ($GA$ and $GA^{(+)}$) as a function of the number of iterations performed. Controller B passively targets all nodes in the network, uniformly, with a budget $B_{B} = N$. Controller A here has only a quarter of the resources available to B, $B_{A} = B_{B}/4 = N/4$. Algorithms are terminated using an approximation factor of $\mu = 10^{-7}$. The learning rate is initialised at $\eta=N$.
• Figure 2: Figure showing relative gain in vote-shares as the distribution of negative edges (p) is varied. Panel (a) shows the effect of heterogeneity of positive (CP-Reg) and negative edges (Reg-CP), compared against networks where both positive and negative components are regular (Reg-Reg). Similarly, panel (b) examines the role of heterogeneity in other commonly studied networks. The heterogeneity of negative edges is varied as the negative graph is changed from a regular (Reg-Reg) to a random (Reg-ER), and finally, a scale-free (Reg-SF) network. Heterogeneity of the positive component is achieved by replacing the regular positive graph (Reg-Reg) with a scale-free network (SF-Reg). For all simulations, networks of size $N=1000$ with $\langle k_{a} \rangle = 16$ and $\langle k_{b} \rangle = 4$ are used. Results are averaged over 10 instances and error bars depict 95% confidence intervals. Note that the panels are scaled differently for clarity.
• Figure 3: Panel showing relative gain in vote-share as $p$ and budget ratios are varied against different competitor strategies. We examine three cases, controller B: (a) avoids nodes with negative ties, (b) targets nodes with negative ties and (c) targets all nodes uniformly. For all cases, controller B has a fixed budget $B_{B}=1$. Lastly, in (d) we quantify gain in vote-share as controller B changes its strategy by varying the fraction of budget $\epsilon_{b}$ used to target nodes with negative edges. Here both controllers have fixed budgets where $B_{A} = 0.3$ and $B_{B}=1$. Results are averaged over 10 CP-Reg-High networks of size $N = 1000$ nodes and $\langle k_{a} \rangle =16$, $\langle k_{b} \rangle = 4$.
• Figure 4: Figures showing (a) the optimal allocations and (b) the vote-shares at equilibrium for values of $p \in [0.075,0.975]$. Results shown for three types of networks : (i) Reg-Reg, (ii) CP-Reg and (iii) Reg-CP. Analytical solutions are shown using dashed lines. Numerical results are obtained through simulations on networks of size N=1000 nodes and averaged over 10 networks. Error bars are shown for $95\%$ confidence intervals.
• Figure 5: Figures showing mean allocations to nodes as a function of competitor allocations $b_{k_{a}k_{b}}$ in SF-Reg networks of size $N=1000$, where $\langle k_{a} \rangle = 50$ and $\langle k_{b} \rangle = 2$. We examine three instances of negative tie distributions $p \in [0.4,0.6,0.8]$ (left to right). Panel (a) shows positive correlations between optimal allocations and competitor allocations and Panel (b) shows negative correlations between optimal allocations and competitor allocations. Different symbols (and colours) correspond to nodes with different negative degrees $k_{b}$. Controllers have the following budgets: (a) $\langle a_{k_{a}k_{b}} \rangle = 5.5$ and $\langle b_{k_{a}k_{b}} \rangle = 0.5$, and (b) $\langle a_{k_{a}k_{b}} \rangle = 4.5$ and $\langle b_{k_{a}k_{b}} \rangle = 5$. Controller B here follows a $k_{a}-$dependent strategy. Numerical results are averaged over 10 networks. Error bars show 95% confidence intervals.