45 problems

Consider real-world networks whose degree distribution follows a power law, and let PageRank be the centrality measure assigned to their vertices. Power-law hypothesis. The PageRan…

Emergent-community conjecture. Communities are an emergent property of local network evolution: networks generated by local rules have finite $r_\kappa$, whereas their degree-prese…

Network-robustness conjecture. The results are robust under the choice of network: other synthetic network models exhibit a subset of the same phases. In particular, non-spatial ne…

The three-level graph ranks fields using theorem-, paper-, and field-level connections, with hyperparameters $(\alpha_T,\alpha_P,\beta_P,\alpha_F)$ controlling the ranking algorith…

Let a thresholded chimpanzee social-interaction network be represented by a graph whose vertices are individuals and whose cliques are sets of individuals with pairwise social conn…

Emergence conjecture. Network communities are an emergent property of networks evolving with local rules.

Let the proportion of people under isolation and the proportion of cities under isolation be measured for the D1+ dataset under strategies $(U)$ and $(P)$, with the initially infec…

Let AFP denote the Annealing with Fixed Points method for approximating graph symmetries, and consider its performance on smaller graphs. Combinatorial-complexity explanation. We c…

Community-membership relevance conjecture. In many empirical networks, the prediction target may be related to whether a node is a strong member of its own community.

Let $\mathcal{G}$ be a network, let $CC(\mathcal{G})$ denote its clustering coefficient, and let $PD_k(\mathcal{G})$ denote its $k$th persistence diagram. Clustering-coefficient co…

The low-density amorphous (LDA) and high-density amorphous (HDA) forms of glassy water are two structural forms observed at low temperatures, where supercooled water remains liquid…

Dependence conjecture. In a network with high reciprocity, nodes with large in- and out-degrees are likely to have highly dependent in- and out-degrees.

Max-iteration conjecture. The max-iteration $k^\infty(G)$ should satisfy the stronger bound suggested by the preceding random-geometric-graph observations.

Let $G$ be a graph with degree sequence $(d_1,\ldots,d_n)$, and let $p_i=d_i/\sum_{j=1}^n d_j$ be its normalized degree sequence. The structural information of $G$ is the Shannon e…

Consider a Watts–Strogatz-type network with high clustering in the coevolutionary opinion-and-action dynamics. The innovation and status quo are the two actions, and a cluster is a…

Let a graph have vertices representing users of a social network, edges representing “friendship”, and weights representing geographical distance. A high-persistence ranging hub is…

Let $G$ be a weakly connected directed graph, and let $\eta_f(G)$ and $\eta_b(G)$ denote its forward and backward democracy coefficients. Democracy coefficient boundedness and bala…

Let $G$ be a directed (unweighted) graph with $m$ edges. Write $eta_f(G)$ and $eta_b(G)$ for its forward and backward democracy coefficients. Democracy coefficient gap conjecture.…

The female connectome is compared with a common female-and-male connectome, and its excess connections are the additional connections present in the female connectome. Excess-conne…

The paper studies multiscale dynamics in adaptive networks, including continuous-time and continuous-state-space systems with a slowly varying network weight. Multiscale network-sc…

Large-agent conjecture. With a large enough number of agents, the EoE will be similar to the EoE in the classic network epidemic model, where network nodes have epidemic states.

The paper considers stochastic block models and possible extensions incorporating overlapping or hierarchical communities, time-dependent or multilayer network structure, metadata,…

Let $cmu_2$ denote the algebraic connectivity of a random geometric graph, let $ckappa_{\mathrm{min}}$ denote its minimum degree, and consider solid random geometric graph ensemble…

Conjecture on unweighted graphs. Introducing jumps always improves the relaxation time: increasing the jump rate from zero decreases the relaxation time for every unweighted graph.

Causal network discovery algorithms evaluate conditional mutual informations over candidate causal parents and conditioning subsets; in dense networks, the number of such tests can…